The Method of Characteristics
The purpose of this chapter is to develop the method of characteristics for the solution of dynamic problems in thermoelasticity. The theoretical analysis of dynamic stresses due to impact loadings has generally been performed by the Laplace transform met
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The Method of Characteristics
The purpose of this chapter is to develop the method of characteristics for the solution of dynamic problems in thermoelasticity. The theoretical analysis of dynamic stresses due to impact loadings has generally been performed by the Laplace transform method. Due to inversion difficulties, the Laplace transform method is usually limited to simple wave problems. The need for the numerical methods to the solution of dynamic problems is dictated by the well-known difficulty of obtaining the exact solutions. Among the various numerical methods, the method of characteristics has the advantages of giving a simple description of the wave fronts and it can give numerical solutions readily to problems with any types of input functions. In mathematics, the method of characteristics is a method of numerical integration of a system of partial differential equations of hyperbolic type. The method is to reduce the hyperbolic partial differential equations to a family of ordinary differential equations, each of which is valid along a different family of characteristic lines (called the characteristics). These equations (called the characteristic equations) are more suitable for numerical analysis because the use of these equations makes it possible to obtain the solutions by a step-by-step integration procedure.
27.1 Basic Equations for Plane Thermoelastic Waves For the shake of motivation, we confine our attention to the solution of onedimensional transient, uncoupled dynamic thermal stresses in plane thermoelastic media subjected to sudden temperature change. The equations which govern the propagation of plane waves in thermoelastic media is given by (1) Equation of motion ρ
∂U ∂σx x = ∂t ∂x
; U=
∂u ∂t
M. Reza Eslami et al., Theory of Elasticity and Thermal Stresses, Solid Mechanics and Its Applications 197, DOI: 10.1007/978-94-007-6356-2_27, © Springer Science+Business Media Dordrecht 2013
(27.1)
715
716
27 The Method of Characteristics
(2) Constitutive equations ∂U ∂T ∂σx x = (λ + 2μ) −β ∂t ∂x ∂t
(27.2)
where x is the Cartesian coordinate and t is the time; u is the displacement and U = ∂u/∂t is the particle velocity in the direction x of the wave propagation; σx x is the normal stress; T is the temperature to be determined independently of the mechanical state of the thermoelastic media; ρ is the density; β = α(3λ + 2μ), where α is the coefficient of thermal expansion; λ and μ are Lamé constants. Eliminating σx x from Eqs. (27.1) and (27.2), we obtain the governing equation in terms of the displacement u ∂2u ρ = 2 ∂x λ + 2μ
∂2u β ∂T + 2 ∂t ρ ∂x
(27.3)
Thus, the dynamic thermoelasticity theory results in the displacement field governed by a hyperbolic second-order partial differential equation, which predicts the finite propagation velocity for mechanical disturbances.
27.2 Characteristics and Characteristic Equations Equation (27.3) is more convenient for the application of the Laplace transform method. For the method of characteristics, we use a system of two linear first
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